Straight-line Generators Research Articles

A Three-Dimensional Generalization of QRT Maps

We propose a geometric construction of three-dimensional birational maps that preserve two pencils of quadrics. The maps act as compositions of involutions, which, in turn, act along the straight line generators of the quadrics of the first pencil and are defined by the intersections with quadrics of the second pencil. On each quadric of the first pencil, the maps act as two-dimensional QRT maps. While these maps are of a pretty high degree in general, we find geometric conditions which guarantee that the degree is reduced to 3. The resulting degree 3 maps are illustrated by two known and two novel Kahan-type discretizations of three-dimensional Nambu systems, including the Euler top and the Zhukovski–Volterra gyrostat with two non-vanishing components of the gyrostatic momentum.

Finding straight line generators through the approximate synthesis of symmetric four-bar coupler curves

The approximate path synthesis of four-bar linkages with symmetric coupler curves is presented. This includes the formulation of a polynomial optimization problem, a characterization of the maximum number of critical points, a complete numerical solution by homotopy continuation, and application to the design of straight line generators. Our approach specifies a desired curve and finds several optimal four-bar linkages with a coupler trace that approximates it. The objective posed simultaneously enforces kinematic accuracy, loop closure, and leads to polynomial first order necessary conditions with a structure that remains the same for any desired trace leading to a generalized result. Ground pivot locations are set as chosen parameters, and it is shown that the objective has a maximum of 73 critical points. The theoretical work is applied to the design of straight line paths. Parameter homotopy runs are executed for 1440 different choices of ground pivots for a thorough exploration. These computations found the expected linkages, namely, Watt, Evans, Roberts, Chebyshev, and other previously unreported linkages which are organized into a 2D atlas using the UMAP algorithm.

Kinematic and kinetostatic analysis of the six-link straight-line generating converting mechanism of the unbalanced sucker rod pumper drive

The results of kinematic and force analysis of the new six-link converting mechanism of the sucker rod pumper drive (SRPD) are presented in this paper and the advantages of the alternative design are substantiated. Using a straight-line generating mechanism allows reducing essentially converting mechanism dimensions and metal consumption as compared with traditionally used SRPD with swinging balancer and crank-based counterweight, first of all, due to eliminating the complicated arc head (so-called «horse-head») of the existing units. However, in order to make sure the working capacities of non-balancer mechanism, kinematic and force characteristics have to be studied. The results of mathematical modeling of the six-link mechanism confirm the qualitative advantages of the straight-line generator. As a result of the study of the rectilinear-guiding mechanism as a transforming mechanism for the drive of sucker rod pumping units, the laws of motion of the links, position, speed and acceleration of all points were determined. To solve the problem of kinematic analysis, the method of closed vector contours was used, which makes it possible to determine the functions of the position of links and analogs of speeds and accelerations. When solving the problem of strength analysis, the equilibrium of each link was considered. As a result of force analysis, jointly solving the equilibrium equations of the links of the six-link hinge-lever mechanism, the reactions of the hinges of the mechanism are determined. A computer model for studying the kinematics and kinetostatics of the converting mechanism of the sucker rod pumping unit drive has been developed

Kempe’s linkages and the Universality Theorem

Inspired by James Watt’s approximate straight line generator, kinematicians of the 19th century challenged themselves to design a mechanical device that could convert rotary motion into a perfect straight line and vice versa. Few inventions emerged in 1864 due to Peaucellier and Lipkin and in 1875 due to Hart. Just a year later, in 1876, Alfred B Kempe presented a generalized method for linkages that could exactly trace any algebraic curve of degree n and not just a straight line. This work of Kempe is of classical importance. Yet, many are not aware of it perhaps because the resulting linkages are quite complex. This article discusses Kempe’s method that highlights the way he treated the rotations analytically using only parallelograms and contra-parallelograms to get the final rigidbody linkage tracing a given algebraic curve. An elaborate example with geometric construction using only a ruler and compass is presented to help the readers understand the assembly of Kempe’s linkages.

Research on Theory and Technique of Point Form Grinding for Quadratic Surface Flank of Twist Drill

A new principle and technology of point form grinding for quadratic surface flank of twist drill was investigated. It broke through the traditional straight-line generator grinding method for the conical and the spiral drills. The principle of implementation mechanism was given in the paper. The trueness and feasibility of the method had been approved in the experiments, and the quadratic surface twist drill such as hyperboloid, ellipsoid and hyperboloid-ellipsoid compound flank had been grinded on the mechanism.

Equilibrium States of Inextensible Membranes under Gravity

We discuss the possible shapes and stress distributions of an inextensible membrane acted on by a gravitational force. It is shown that there are smooth shapes which cannot be an equilibrium solution under gravity for any stress distribution. In addition there are infinitely many shapes which are equilibrium solutions under gravity and for which the stress distribution is unique. Finally, there are infinitely many solution surfaces – each consisting of straight line generators parallel to the direction of gravity. These solutions are possible under infinitely many different stress distributions. In several cases we will solve the equations explicitly to find actual shapes and stress distributions.

Three-Dimensional Vibration Analysis of Thick Shells of Revolution

A 3D method of analysis is presented for determining the free vibration frequencies and mode shapes of hollow bodies of revolution (i.e., thick shells), not limited to straight-line generators or constant thickness. The middle surface of the shell may have arbitrary curvatures, and the wall thickness may vary arbitrarily. Displacement components u;gf, uz, and u;gu in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the ϕ and z directions. Potential (strain) and kinetic energies of the entire body are formulated, and upper-bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials are increased, frequencies converge to the exact values. Novel numerical results are presented for two types of thick conical shells and thick spherical shell segments having linear thickness variations and completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of both types of shells. The method is applicable to thin shells, as well as thick and very thick ones.

Flow initiation in loose frictional solids

Isotropic, loose frictional solids possess yield surfaces with straight line generators which form regular hexagons when viewed in the octahedral direction.

Design of transition sections between ducts of equal area

The geometry of transition pipe sections with the same cross-sectional area at both ends, but varying in cross-sectional shape, has been studied by Atilgan and Calvert (190). The transition pipes studied were designed using straight-line generators. They included transitions between rectangular and rectangular, and between circular and rectangular ducts. They had an area variation of parabolic form with the maximum occuring at the mid-point. This report presents a new design for these transition pipes to provide constant cross-sectional area along the transitions. Transitions between rectangular or circular and rectangular ducts are considered. The rectangular to rectangular transitions are designed to have either one pair of flat walls and one pair of curved walls, or all wall curved. The circular to rectangular transition pipes are made up of circular quadrants at the corners with one pair of flat walls and one pair of curved walls, all walls curved, or all walls flat. In all cases, the degree of wall curvature is only a function of the aspect ratio at the rectangular end.

Geometry of transitional diffusers

Practical transitional diffuser geometries with and without varying cross-sectional shape are studied. Transitional diffusers between rectangular and rectangular, circular and rectangular, circular and equilateral triangular, and between rectangular and rectangular with offset centre-lines are investigated. The analysis presented is based on straight-line generators along the transitional diffusers in all cases. The analysis shows that the amount of area variation depends on the aspect ratio of the rectangular section. For straight transitional diffusers between rectangular and rectangular and between circular and rectangular or triangular, the area variation is of parabolic form with either a relative reduction or a relative increase of area at the midpoint. Transitional diffusers, which provide uniform area distributions, may also be obtained. The area variation becomes more complicated when the area perpendicular to the transition centre-line is considered, in transitions with the inlet and outlet duct centre-lines offset. Here, for a rectangular-to-rectangular transitional diffuser, the variation depends on both the aspect ratios and the angles of offset.

Area distribution along general transition geometries

General transition geometries with the same area at both ends, but of varying cross-sectional shape are studied. Transitions between circular and polygon, rectangular and regular polygon, and between rectangular and rectangular with offset centre lines are investigated. In all cases, the analysis is based on straight-line generators along the transitions. The analyses show that the amount of area variation depends on the number of sides of the polygonal section. This variation is of parabolic form along the transition, with the maximum at the mid-point. However, the variation becomes more complicated when the area perpendicular to the transition centre line is considered, in transitions with the inlet and outlet duct centre lines offset. Here, for a rectangular-to-rectangular transition, the variation depends on both the aspect ratios and the angle of offset.

Optimization of parameters for specified path generation using an atlas of coupler curves of geared five-bar linkages

An atlas containing 732 coupler curves traced by the symmetric geared five-bar linkage has been drawn by the authors. Gear ratios of +1, −1, +2, −2, have been used. The data are presented either six or twelve curves to a page, arranged in axonometric projection to illustrate the envelope of a family of curves of that particular linkage geometry as the gearset phase angle isvaried. Dots on the curve represent uniform intervals of input crank angular displacement in order to provide coupler point velocity information. Other curves for parameters lying between those illustrated can be visually interpolated. Optimization techniques can be applied to modify the geometrics of a linkage parameters selected from the atlas as an approximate solution to the path motion desired. The origincurves are drawn on 11 × 17 sheets in four colors with a Benson-Varian xy plotter. Optimization algorithms are then used in conjunction with the atlas of linkage coupler curves to refine an initial selection of a desired motion path to achieve a particular motion with minimum deviation. The geared five-bar linkage is used as an example and a straight line generator is designed with less than 1% position error.

Sensitivity analysis of the inverted slider crank straight-line generator

This paper presents the results of a study of an inverted slider crank straight-line generator and its sensitivity to link-length tolerances. Analytical results are presented graphically using a Tektronix CRT to assist the designer in making intelligent choices for tolerancing the links, based on his performance requirements.

Geometry of transition sections between ducts of equal area

The report presents an analysis of the geometry of transition pipe sections with the same area at both ends, but varying cross-sectional shape. Transitions between rectangular and rectangular, and circular and rectangular are investigated. Straight-line generators are used along the transitions in both cases. An equivalent cone angle concept is used as a design parameter. The cross-sectional area varies considerably along the transition sections. In all cases, the area variation is of quadratic form with the maximum area occurring at the centre and being a function of aspect ratios only. The area change increases as aspect ration falls. Due to change in the area, the equivalent cone angle decreases to zero at the centre of the transition and thereafter becomes negative for any duct length. The equivalent cone angle at entry depends only on aspect ratios and overall non-dimensionalised lengths. With increasing duct length, the maximum equivalent cone angle decreases considerably. Generally, circular-to-rectangular transitions produce lower area variations than rectangular-to-rectangular transitions for the same aspect ratios. Square-to-rectangular transitions have the least variation. It is suggested that the overall pressure losses along transition sections can be calculated from knowledge of flow through diffusers and contractions.

Axisymmetric transients in shells of revolution

AbstractThe propagation of axisymmetric transients in shells of revolution subjected to impulsive boundary loads is treated in this paper. Consideration is restricted to linear, elastic, isotropic, homogeneous shells of revolution with straight line generators. The analysis is based on the concept of a wave as a carrier of discontinuities in the field variable and its derivatives. These discontinuities are determined from a set of recurrence relations which are in turn generated by the use of asymptotic series solutions to the equations of motion. A numerical superposition technique which enables the calculation of long time response is developed. Several numerical examples are presented in order to illustrate the method.

A Dynamic Synthesis of Non-uniform Motion Mechanisms

Taking link mechanisms as an example of non-uniform motion mechanisms, a dynamic synthesis has been carried out ; dynamic parameters such as masses and spring constants of links have been determined together with kinematic parameters such as length of links such that the errors of outputs at the prescribed performance speeds might become minimum. The synthesis has been applied to a customary four-bar straight-line generator yielding a straight line longer by 52% than the original one. The internal forces and deformations of links and the deviations of outputs caused by the external forces have been also analyzed on general link mechanisms with a single loop.